Why do local sidereal time and apparent solar time coincide at autumnal equinox?

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astrostudent

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According to my notes, local sidereal time is the hour angle of the first point of Aries. And apparent solar time is the hour angle of the sun. So at the vernal equinox, when the sun is at the first point of Aries, surely the HA of the sun equals the HA of the first point of Aries, and so LST = AST.

But various websites say that LST is the same as AST at the autumnal equinox.

Can anyone explain why this is?
 
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